We can model the synovial fluid in an artificial hip as a Newtonian fluid between two curved surfaces, with radii R₁ and R₂, as shown in Figure Q1. We can treat the motion as being steady, with the outer surface being stationary and the inner surface moving at a speed wR₁. We can assume the pressure, p, is constant everywhere and that the velocity (ug and u,) does not change in the direction. ↑ R₂ WR, R₁ Ue(r) Figure Q1: Synovial fluid in a hip joint. We can describe the flow of fluid in the joint using the two-dimensional Navier-Stokes equations in cylindrical coordinates (i.e. we neglect the velocity and any velocity gradients in the z direction). The component of the 2D Navier-Stokes equation is див Ив див 1 др P + Ur = at ((()) + + +) [Eqn 1] 202 т дв And the 2D Continuity Equation is given by r ar 1ǝ(rur) 1 див + т де = 0 [Eqn 2] a) Using the Continuity Equation (Eqn 2), find the radial velocity, u,, in the fluid. [8 marks] b) Simplify the Navier-Stokes (Eqn 1), clearly stating any assumptions you make c) Find an expression for the azimuthal velocity, up throughout the fluid [12 marks] [10 marks]

We can model the synovial fluid in an artificial hip as a Newtonian fluid between two curved surfaces, with radii R₁ and R₂, as shown in Figure Q1. We can treat the motion as being steady, with the outer surface being stationary and the inner surface moving at a speed wR₁. We can assume the pressure, p, is constant everywhere and that the velocity (ug and u,) does not change in the direction. ↑ R₂ WR, R₁ Ue(r) Figure Q1: Synovial fluid in a hip joint. We can describe the flow of fluid in the joint using the two-dimensional Navier-Stokes equations in cylindrical coordinates (i.e. we neglect the velocity and any velocity gradients in the z direction). The component of the 2D Navier-Stokes equation is див Ив див 1 др P + Ur = at ((()) + + +) [Eqn 1] 202 т дв And the 2D Continuity Equation is given by r ar 1ǝ(rur) 1 див + т де = 0 [Eqn 2] a) Using the Continuity Equation (Eqn 2), find the radial velocity, u,, in the fluid. [8 marks] b) Simplify the Navier-Stokes (Eqn 1), clearly stating any assumptions you make c) Find an expression for the azimuthal velocity, up throughout the fluid [12 marks] [10 marks]